Definition of Functions

Q.

Express $\tan \theta$ in terms of $\sin \theta$.

Q. $Ifx=\sqrt{1+\sqrt{1+\sqrt{1+......toinfinity}}},thenx=$

1. 
2. 
3. 
4. None of these

Q.

How do you express $\sin \left(3\theta \right)$ in terms of trigonometric functions of $\theta$?

Q.

Can you find the square root of a negative number? Explain.

Q.

If $A=[x:xisamultipleof3]andB=[x:xisamultipleof5],$ then $A–B$is

Q.

Name The Type Of Quadrilateral Formed By The Following Points -$(-1,-2),(1,0),(-1,2),(-3,0)$ And Give Reasons For Your Answer

Q.

The real values of $\mathrm{x}$ and $y$ for which the equation is $(\mathrm{x}+\mathrm{iy})(2-3\mathrm{i})=4+\mathrm{i}$ is satisfied, are

1. $x=\frac{5}{13},y=\frac{8}{13}$

2. $x=\frac{8}{13},y=\frac{5}{13}$

3. $x=\frac{5}{13},y=\frac{14}{13}$

4. None of these

Q.

Which of the following is a singleton set?

1. $\{x:|x|=5,x\in \mathrm{\mathbb{N}}\}$

2. $\{x:|x|=6,x\in \mathrm{\mathbb{Z}}\}$

3. $\left\{x:x^2+2x+1=0,x\in \mathrm{\mathbb{N}}\right\}$

4. $\left\{x:x^2=7,x\in \mathrm{\mathbb{N}}\right\}$

Q. Which of the following is/are function(s)?

1. 
2. 
3. 
4. 

Q. Let $A=\{a,b,c\}$ and $B=\{1,2,3,4\}$. Then the number of elements in the set $C=\{f:A\to B|2\in f(A\left)\text{and}f\text{is not one–one}\right\}$ is

Q. Two clocks are set to show $4\text{am}$ on ${21}^{\text{st}}$ November $2021.$ One clock gains $5$ minutes in an hour and the other clock loses $2$ minutes in an hour. Then by how many minutes do the two clocks differ at $8\text{pm}$ on ${22}^{\text{nd}}$ November $2021$?

1. $280$ minutes
2. $120$ minutes
3. $112$ minutes
4. $266$ minutes

Q.

The domain of the function $f\left(x\right)={\log }_{3+x}\left(x^2-1\right)$ is

1. $\left(-3,-1\right)\cup \left(1,\infty \right)$

2. $[-3,-1)\cup [1,\infty )$

3. $\left(-3,-2\right)\cup \left(-2,-1\right)\cup \left(1,\infty \right)$

4. $[-3,-1)\cup \left(-2,-1\right)\cup [1,\infty )$

Q.

Solve for x :

$6^x+4^x=9^x$

Q.

Jagdish can build a wall in $10$ days. Narender can build the same wall in $12$ days while Sumit takes $15$ days to do the same job. Which two of them should be employed to finish the job in $6$ days?

Q.

What is a periodic matrix?

Q. Let $f\left(x\right)=x^{13}+x^{11}+x^9+x^7+x^5+x^3+x+12.$ Then

1. $f\left(x\right)$ has $13$ non-zero real roots
2. $f\left(x\right)$ has exactly one real root
3. $f\left(x\right)$ has exactly one pair of imaginary roots
4. $f\left(x\right)$ has no real root

Q.

Is $x^3-4x^2-x+1={(x-2)}^3$ a quadratic equation?

Q.

The number of solutions of the equation ${\log }_4\left(x-1\right)={\log }_2\left(x-3\right)$ is

Q. The number of points where the function $f\left(x\right)=\{\begin{array}{ccc}|2x^2-3x-7|,& \text{if}& x\le -1\\ [4x^2-1],& \text{if}& -1<x<1\\ |x+1|+|x-2|& \text{if},& x\ge 1\end{array},$ where $\left[t\right]$ denotes the greatest integer $\le t$, is discontinuous is

Q.

If $a=\cos \frac{2\pi }{7}+i\sin \frac{2\pi }{7}$ then the quadratic equation whose roots are $\alpha =a+a^2+a^4$ and $\beta =a^3+a^5+a^6$is

1. $x^2-x+2$

2. $x^2+x-2$

3. $x^2-x-2$

4. $x^2+x+2$

Q. If $f\left(x\right)=4x-x^2,x\in R$, then $f(a+1)-f(a-1)$ is equal to

1. $2(4-a)$
2. $4(2-a)$
3. $4(2+a)$
4. $2(4+a)$

Q. Find the domain and range of the function $f\left(x\right)=\frac{1}{\sqrt{x-5}}$

Q.

What is the difference between real and complex roots?

Q. Suppose that $a,b,x$ and $y$ are real numbers such that $ax+by=3,$ $a{x}^2+b{y}^2=7,$ $a{x}^3+b{y}^3=16$ and $a{x}^4+b{y}^4=42.$ Find the value of $a{x}^5+b{y}^5.$
(correct answer + 3, wrong answer 0)

Q.

Can the quadratic polynomial${\mathrm{x}}^2+\mathrm{kx}+\mathrm{k}$have equal zeroes for some odd integer $\mathrm{k}>1$?

Q. For a function $f:A\to B$, the number of elements in set $A$ and $B$ are $5$ and $4$ respectively, then the number of many one functions are

Q.

Let A =R -{3} and B=R -{1}. Consider the function $f:A\to B$ defined by $f\left(x\right)=\left(\frac{x-2}{x-3}\right)$ is f one-one and onto? Justify your answer.

Q.

Are square root functions even or odd?

Q.

The minimum value of $\frac{x}{\log \left(x\right)}$ is

1. $e$

2. $\frac{1}{e}$

3. $e^2$

4. $e^3$

Q.

(C₀ / 1) + (C₂ / 3) + (C₄ / 5) + (C₆ / 7) + ….. = +

1. $\frac{2^{n+1}}{(n+1)}$

2. $\frac{2^{n+1}-1}{n+1}$

3. $\frac{2^n}{(n+1)}$

4. None of these